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<title>Atlas software user guide -- Scope</title>
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<h2>Scope of the program</h2>
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<i>Last updated: October 8, 2005</i>
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The groups considered in this program are the full groups of real points
of connected reductive complex algebraic groups defined over <b>R</b>. In
particular, these groups are often non-connected in the analytic topology.
It should not be hard at some point to allow for open subgroups of such groups
(e.g., their identity components) to be considered. A much more challenging
extension would be to allow for finite covers of such open subgroups (i.e.,
non-linear real reductive groups); currently, this is not envisioned at all.
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<p>
Here is how a real reductive group is input interactively in the program:
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<li>
We choose the Lie algebra <b>g</b> of our connected complex reductive 
group. This is done by giving the <i>Lie type</i> of <b>g</b> (a sequence of 
symbols like A3.E6.T2, where
the "T" factors indicate torus factors&#8212;for instance, T2 indicates a torus
factor of rank 2). The only constraint here is that the total rank should
not exceed 16 (this can be easily increased if your machine is powerful enough,
but computations for bigger groups become quickly unfeasible anyway.)
</li>
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<li>
A group G with Lie algebra <b>g</b> is chosen. For this, we start out
with the group that is the direct product of the simply connected group
corresponding to the semisimple part of <b>g</b> and a torus of the appropriate
rank, and mod out by a finite subgroup of the center. What the user must do
is enter generators for the central subgroup he wants to mod out by.
</li>
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<li>
We need to choose an <a href="innerclass.html">inner class</a> of real forms
for G. In practice, an inner class is encoded by a string of letters like 
&#8220;cscCu&#8221;, where c stands for compact, s for split, C for complex and
u for &#8220;unequal rank&#8221;. Each inner class symbol applies to the
corresponding entry in the Lie type sequence, except that a complex inner class
consumes two Lie type symbols (that must be equal.) The u symbol is the only
way to access the non-equal rank inner class in <a href="Deven.html"> 
type D<sub>2m</sub></a>; for the other types that have two inner classes 
(A<sub>n</sub>, n > 1, D<sub>2m+1</sub> and E<sub>6</sub>), u is the same as s.
For consistency, e (&#8220;equal rank&#8221;) is also allowed; this is the same
as c.
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<li>
Finally, if and when it is required to actually specify a real form, the
program will compute the classification of weak real forms for the current
inner class, and present them to the user through the corresponding real forms
of the Lie algebra <b>g</b>. For example, if the Lie type is A3, and the
inner class is "c", the choices will be su(4), su(3,1), and su(2,2). It turns
out that when <b>g</b> is simple, the isomorphism class of the real form of
<b>g</b> entirely determines the weak real form of G, except again in
<a href="Deven.html">type D<sub>2m</sub></a></a>. For general semisimple or 
reductive groups, this will no longer be true.
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The choice of a real form as in 4 may actually not be required at all for a
number of commands; or it may be triggered only much later in the session,
when the real form actually matters. It is also possible to switch from
one real form to the other within the same inner class, without changing the 
underlying G (and preserving whatever computations had already been done for G)
through the "realform" command.
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<p>
<a href="userintro.html">Back</a> to the introduction.<br>
<a href="http://atlas.math.umd.edu">Back</a> to the Atlas homepage.
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